Maximum with Lagrange multiplier

I am trying to find the maximum of f(x,y)=(x+y)4+y4f(x,y)=(x+y)^4+y^4 constrained to x4+y4=1x^4+y^4=1.

Using Lagrange Multiplier I get
(x+y)3=λx3
(x+y)^3=\lambda x^3

(x+y)3+y3=λy3
(x+y)^3+y^3=\lambda y^3

But I don’t see how to proceed after this.

Do you have some idea on this problem ?

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Is f(x,y)=x+y4+y4=x+2y4f(x,y)=x+y^4+y^4=x+2y^4 right or do you have a typo ?
– callculus
2 days ago

  

 

Thank you. I have corrected the formula.
– Spout
2 days ago

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1 Answer
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I don’t see a simple way to proceed from here, but the basic idea is that you now have three equations — the two equations you derived and the constraint equation x4+y4=1x^4 + y^4 =1 — and three unknowns xx, yy, and λ\lambda. You then combine the three equations to solve for xx and yy. (And you can also solve for λ\lambda but that is just gravy.)